Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{-2z^3 - 20z^2 - 42z}{9z^2 + 54z + 81}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {-2z(z^2 + 10z + 21)} {9(z^2 + 6z + 9)} $ $ r = -\dfrac{2z}{9} \cdot \dfrac{z^2 + 10z + 21}{z^2 + 6z + 9} $ Next factor the numerator and denominator. $ r = - \dfrac{2z}{9} \cdot \dfrac{(z + 3)(z + 7)}{(z + 3)(z + 3)}$ Assuming $z \neq -3$ , we can cancel the $z + 3$ $ r = - \dfrac{2z}{9} \cdot \dfrac{z + 7}{z + 3}$ Therefore: $ r = \dfrac{ -2z(z + 7)}{ 9(z + 3)}$, $z \neq -3$